Precalculus Examples

$\frac{1}{2} \cdot (\log_{7} (x) + \log_{7} (y) - 6 \log_{7} (x + 6))$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\frac{1}{2} \cdot (\log_{7} (x y) - 6 \log_{7} (x + 6))$

Step 2

Simplify $- 6 \log_{7} (x + 6)$ by moving $6$ inside the logarithm.

$\frac{1}{2} \cdot (\log_{7} (x y) - \log_{7} ({(x + 6)}^{6}))$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1}{2} \cdot \log_{7} (\frac{x y}{{(x + 6)}^{6}})$

Step 4

Simplify $\frac{1}{2} \log_{7} (\frac{x y}{{(x + 6)}^{6}})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{7} ({(\frac{x y}{{(x + 6)}^{6}})}^{\frac{1}{2}})$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 5.1

Apply the product rule to $\frac{x y}{{(x + 6)}^{6}}$ .

$\log_{7} (\frac{{(x y)}^{\frac{1}{2}}}{{({(x + 6)}^{6})}^{\frac{1}{2}}})$

Step 5.2

Apply the product rule to $x y$ .

$\log_{7} (\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{{({(x + 6)}^{6})}^{\frac{1}{2}}})$

Step 6

Multiply the exponents in ${({(x + 6)}^{6})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{7} (\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{{(x + 6)}^{6 (\frac{1}{2})}})$

Step 6.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1

Factor $2$ out of $6$ .

$\log_{7} (\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{{(x + 6)}^{2 (3) \frac{1}{2}}})$

Step 6.2.2

Cancel the common factor.

$\log_{7} (\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{{(x + 6)}^{2 \cdot 3 \frac{1}{2}}})$

Step 6.2.3

Rewrite the expression.

$\log_{7} (\frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{{(x + 6)}^{3}})$